Extended Kalman Filter (EKF) Experimental Report

Extended Kalman Filter (EKF) Experimental Report

The Extended Kalman Filter (EKF) is a powerful algorithm for state estimation in nonlinear systems, widely applied in robotics localization, target tracking, and sensor fusion. Compared to the standard Kalman Filter (KF), EKF linearizes nonlinear systems through first-order Taylor expansion, thereby preserving KF's efficient recursive characteristics.

Experimental Objectives This experiment primarily validates EKF's state estimation capability in nonlinear systems. Through comparative analysis of simulated data and actual sensor data, we evaluate its estimation accuracy and stability. The experiment includes system modeling, Jacobian matrix computation, implementation of prediction and update steps, and filtering performance assessment.

Experimental Methodology System Modeling: Establish state transition and measurement equations for nonlinear motion or observation models (e.g., robot kinematics or sensor observation equations) Linearization Processing: Compute Jacobian matrices for state transition and observation matrices at each iteration to achieve local linearization Prediction and Update: Prediction Phase: Calculate prior estimates based on previous state and covariance matrices Update Phase: Combine sensor measurements to correct prior estimates and obtain posterior estimates Performance Evaluation: Quantify EKF's estimation accuracy using Root Mean Square Error (RMSE) or comparison withground-truth trajectories

Key Implementation Details: - System modeling typically involves defining state vectors (position, velocity) and nonlinear functions for state transitions - Jacobian calculation requires partial derivatives of system models w.r.t state variables - Prediction step uses: x_prior = f(x_prev), P_prior = F*P_prev*F' + Q - Update step applies: K = P_prior*H'*(H*P_prior*H' + R)^-1, x_post = x_prior + K*(z - h(x_prior))

Experimental Results The experiment generates state trajectories of nonlinear systems through simulation (e.g., motion models with turning maneuvers) and adds Gaussian noise to simulate sensor errors. EKF effectively tracks true states, particularly excelling under weak nonlinear conditions. However, under strong nonlinearities or large initial errors, filter divergence may occur due to linearization errors. In such cases, improved algorithms like Unscented Kalman Filter (UKF) can be considered.

Implementation Considerations: - For strong nonlinearities, UKF uses sigma points instead of Jacobians for better approximation - Code implementation should include checks for matrix positive definiteness and numerical stability - Real-time applications require optimized Jacobian computation methods

Conclusion EKF provides a feasible real-time state estimation solution for nonlinear systems, with its core depending on the reasonableness of local linearization. Experiments demonstrate that proper parameter tuning (e.g., process noise and observation noise covariance) is crucial for stability. Future work could explore adaptive EKF or hybrid filtering methods to enhance robustness.

Important Note: Practical applications must consider computational efficiency and numerical stability in Jacobian matrix calculations to prevent filter failure due to accumulated linearization errors. Code implementations should include: - Efficient symbolic/numerical differentiation for Jacobians - Numerical safeguards against ill-conditioned matrices - Regular covariance matrix checks and resets if needed